map(Module,Module,RingElement) -- construct the map induced by multiplication by a ring element on the generators

Synopsis

Function: map
Usage:

map(M,N,r)
Inputs:
- M, a module
- N, a module, over the same ring R as M. An integer here stands for the free module of that rank.
- r, a ring element, in the ring R
Optional inputs:
- Degree => ..., default value null, specify the degree of a map
- DegreeLift => ..., default value null, make a ring map
- DegreeMap => ..., default value null, make a ring map
Outputs:
- a matrix, The map induced by multiplication by r on the generators

Description

If r is not zero, then either M and N should be equal, or they should have the same number of generators. This gives the same map as r * map(M,N,1). map(M,N,1) is the map induced by the identity on the generators of M and N.

i1 : R = QQ[x];

i2 : map(R^2,R^3,0)

o2 = 0

             2      3
o2 : Matrix R  <-- R

i3 : f = map(R^2,R^2,x)

o3 = | x 0 |
     | 0 x |

             2      2
o3 : Matrix R  <-- R

i4 : f == x *map(R^2,R^2,1)

o4 = true

Caveat

If M or N is not free, then we do not check that the result is a well defined homomorphism.

Ways to use this method: